Two graphs $G$ and $H$ are homomorphism indistinguishable over a class of graphs $\mathcal{F}$ if for all graphs $F \in \mathcal{F}$ the number of homomorphisms from $F$ to $G$ is equal to the number of homomorphisms from $F$ to $H$. Many natural equivalence relations comparing graphs such as (quantum) isomorphism, spectral, and logical equivalences can be characterised as homomorphism indistinguishability relations over certain graph classes. Abstracting from the wealth of such instances, we show in this paper that equivalences w.r.t. any self-complementarity logic admitting a characterisation as homomorphism indistinguishability relation can be characterised by homomorphism indistinguishability over a minor-closed graph class. Self-complementarity is a mild property satisfied by most well-studied logics. This result follows from a correspondence between closure properties of a graph class and preservation properties of its homomorphism indistinguishability relation. Furthermore, we classify all graph classes which are in a sense finite (essentially profinite) and satisfy the maximality condition of being homomorphism distinguishing closed, i.e. adding any graph to the class strictly refines its homomorphism indistinguishability relation. Thereby, we answer various questions raised by Roberson (2022) on general properties of the homomorphism distinguishing closure.

翻译：两个图 $G$ 和 $H$ 在某个图类 $\mathcal{F}$ 上同态不可区分，是指对于所有图 $F \in \mathcal{F}$，从 $F$ 到 $G$ 的同态数量等于从 $F$ 到 $H$ 的同态数量。许多用于比较图的自然等价关系，如（量子）同构、谱等价和逻辑等价性，均可刻画为特定图类上的同态不可区分关系。基于这些丰富的实例，本文证明：任何满足自互补性且可刻画为同态不可区分关系的逻辑等价关系，均可通过某个禁子式封闭图类上的同态不可区分性来刻画。自互补性是一个温和的性质，大多数被广泛研究的逻辑均满足该性质。这一结果来源于图类的闭包性质与其同态不可区分关系的保持性质之间的对应关系。此外，我们分类了所有在某种意义下有限（本质上为射有限）且满足极大性条件（即同态不可区分闭集）的图类，其中向类中添加任意图都会严格细化其同态不可区分关系。由此，我们回答了Roberson（2022）关于同态不可区分闭包一般性质提出的多个问题。